Simplicity is the understanding that God is not composed of parts. There are no attributes or generic nature lying around which when combined in the right way produce God, like a recipe produces a cake.

First, God’s existence (act of being) and essence (quiddity) cannot be constituent components in Him, each supplying what the other lacks. Rather, God must be identical with His existence and essence, and they must be identical with each other. It is His essence to be. Strictly speaking, His act of existence is not what He has, but what He is.

Dolezal, James E.. All That Is in God: Evangelical Theology and the Challenge of Classical Christian Theism . Reformation Heritage Books. Kindle Edition.

A second aspect of simplicity guards against dividing God’s attributes into separate things — parts of God:

Now Christian theology has always been more or less conscious of this calling. On the whole, its teaching has been that God is “simple,” that is, sublimely free from all composition, and that therefore one cannot make any real [i.e., ontological] distinction between his being and his attributes. Each attribute is identical with God’s being: he is what he possesses. In speaking of creatures we make all sorts of distinctions between what they are and what they have. A person, for example, is still human even though he or she has lost the image of God and has become a sinner. But in God all his attributes are identical with his being. God is light through and through; he is all mind, all wisdom, all logos, all spirit, and so forth.67 In God “to be is the same as to be wise, which is the same as to be good, which is the same as to be powerful. One and the same thing is stated whether it be said that God is eternal or immortal or good or just.” Whatever God is, he is that completely and simultaneously. “God has no properties but is pure essence. God’s properties are really the same as his essence: they neither differ from his essence nor do they differ materially from each other.”

Herman Bavinck, John Bolt, and John Vriend, Reformed Dogmatics: God and Creation, vol. 2 (Grand Rapids, MI: Baker Academic, 2004), 118.  These are admittedly difficult things to keep in mind — because this is not how our world exists.

Creatures are created things — they exist because they were composed, built by God.  But such segregation and separation of parts became more extraordinary with the entrance of death:

What then is spiritual death? Of course it entails severing the bond that God created in us at creation, but which bond? The answer is: the spiritual bond that connects our soul with God. Not only our body is tied to our soul with a bond, but [at creation] our soul was also tied with a bond to God. That bond is automatically unraveled through sin, and thus immediately at this point death enters simultaneously with sin. Instead of drinking in life with God, the soul is thrown back upon itself, even as a pipe unscrewed from the water supply empties out and dries up. It is thus entirely understandable that there is a dying, a death, in two respects. One involves the tearing asunder of the bond between body and soul in us, the other is a dying in which the bond between the soul and God is torn apart.

Abraham Kuyper, Common Grace: God’s Gifts for a Fallen World: The Historical Section, ed. Jordan J. Ballor, Melvin Flikkema, and Stephen J. Grabill, trans. Nelson D. Kloosterman and Ed M. van der Maas, vol. 1, Abraham Kuyper Collected Works in Public Theology (Bellingham, WA: Lexham Press; Acton Institute, 2015), 247.

Hence being a creature and living in a world which decays into parts makes the concept of a simple God very difficult.

This difficulty seems acute when we come to something such as the Trinity. How is that a simple God could be one God and three Persons? The obvious answer is to try to divide God into three Persons and then try to compose something which have sufficient interaction to make some sort of a “one”.

Yet, a division into parts, indeed into three gods, is unacceptable if we are to take the Scripture seriously. The New Testament, which more fully discloses the Trinity, does not lessen the absolute unity of the One God (indeed, this is one of the things which makes the early Church’s veneration of Jesus as God so striking — how indeed could these early Christians have believed in One God, One Father and One Son — not to mention One Spirit — all at once). Christianity cannot maintain its integrity and permit any division of God into any parts:

To affirm God’s spirituality is also to affirm his simplicity. Christian faith is adamant that God is one and indivisible, that he does not encompass within himself disparate parts or quantities.

Donald G. Bloesch, God, the Almighty: Power, Wisdom, Holiness, Love (Downers Grove, IL: InterVarsity Press, 2006), 90.

If we divide God into Gods, if we try to somehow lessen the simplicity of God to better make sense of the Trinity — to our thinking — we end up creating something which is an addition to God. The very act of trying to find divisions of being in the Godhead, to make the Trinity more easily comprehensible, will create something extra to God which is necessary for the God to be God (and what could such a thing be?):

By reason of its incomplexity and simplicity, divine essence is indivisible. Not being made up, as matter is, of diverse parts or properties, it cannot be divided or analyzed into them: “The nature of the Trinity is denominated simple, because it has not anything which it can lose and because it is not one thing and its contents another, as a cup and the liquor, or a body and its color, or the air and the light and heat of it” (Augustine, City of God 11.10).

William Greenough Thayer Shedd, Dogmatic Theology, ed. Alan W. Gomes, 3rd ed. (Phillipsburg, NJ: P & R Pub., 2003), 223. The divided parts would be something not-God.

Here is where are thinking must be precise — and precisely where it is most difficult. If we were to think of individual persons who were human beings, we would think of human nature and then human beings. They would be divided by place and appearance and whatnot:

In a multitude of beings of the same kind or class there is something more in the being of the individual than just the nature or essence by which it is defined. That is, something more than the nature or essence as such gives it distinction from all others in the class. This distinctive quality may be one’s particular matter or perhaps some other accidental features of its being.

Dolezal. Location in time and space are something which exist independently of human nature and permit us to distinguish one person from another. One man lived in New York in 1900 another man lived in Los Angeles in 2000. That time and space is an accident which is coupled to human nature and distinguish the two men (there would be numerous accidents which could be used to distinguish both men). Those distinguishing marks are things which can be separated from human nature while the human nature remains.

Yet, as we have seen, if we were to distinguish the members of the Trinity in the same way, we would draw on something outside of God to add to the Son or the Father, some “particularizing feature” which would not be God to distinguish God from God:

But in God, there can be nothing that He is that lies outside His nature—no determination of His being in addition to His essence. If there were, God would require something beyond His divinity, His Godness, for the fullness of His being. For God to be divine and for God to be this God we call Yahweh are one and the same reality. Thus, divinity cannot be a genus or species in which divine persons exist as so many particular instantiations.

Those who maintain the classical doctrine of simplicity deny that there is any distinction in God between suppositum and nature. God has no real particularizing features over and above His divine nature. This feature of simplicity rules out any possibility that true divinity could appear in a plurality of beings really distinct from each other, for instance, as true humanity (nature/essence) is able to appear in a plurality of really distinct humans (supposita). It is thus divine simplicity that undergirds monotheism and ensures that it does not just so happen that God is one, but it must be that God cannot but be one being because of what it means to be God.

Dolezal, James E.. All That Is in God: Evangelical Theology and the Challenge of Classical Christian Theism . Reformation Heritage Books. Kindle Edition.

How then to we maintain the simplicity of God and the Trinity? The Trinity is how this one God is:

What, then, are we saying about God when we speak of the Father, Son, and Holy Spirit? First, it should be observed that we are not speaking of things that are distinct from the Godhead itself. Whenever we speak of the three, we are in fact speaking of the one, but under different aspects or modes of being. We alternatively speak of the one God Father-wise, Son-wise, and Spirit-wise—in sum, relation-wise. These relations are not something really distinct from the divine substance. As John Owen puts it, “A divine person is nothing but the divine essence…subsisting in an especial manner.”37 The challenge is that in our creaturely experience our talk about substances and our talk about relations must necessarily be distinguished. When we speak of what belongs to humans as human, we speak of them according to substance. When we speak of them as a parent, child, friend, employee, and so forth, we speak according to relation. Because these two realities—substance and relation—are not strictly identical in the human subject, we speak of them as really distinct features of the human’s being. Indeed, we have no other speech pattern available to us. But in God, relations are not features of His being that exist over and above His substance. They add nothing to the substance. They are not principles of actuality adjoined to the divine essence that determine it to exist in some sense, as if the essence were something abstract that is then made concrete in the persons. In God, there is no mixture of abstract and concrete. We are forced to speak of God’s essence under the rubric of substance terminology and relation terminology, which Augustine calls “substance-wise” and “relationship-wise.”38 Our inability to say or even think both at once is why we must proceed in this double way of speaking of the one God.39 Yet this double way of speaking of God, alternatively according to substance and relation, is not to be understood to mirror a double way of being within Himself. He is not composed of substance and relations as creatures are.

Dolezal, James E.. All That Is in God: Evangelical Theology and the Challenge of Classical Christian Theism . Reformation Heritage Books. Kindle Edition.

Dolezal quotes Owen in brief, here is the entire paragraph. And in what might be the only instance in Western Civilization, a quotation from John Owen may be clarifying:

The distinction which the Scripture reveals between Father, Son, and Spirit, is that whereby they are three hypostases or persons, distinctly subsisting in the same divine essence or being. Now, a divine person is nothing but the divine essence, upon the account of an especial property, subsisting in an especial manner. As in the person of the Father there is the divine essence and being, with its property of begetting the Son, subsisting in an especial manner as the Father, and because this person has the whole divine nature, all the essential properties of that nature are in that person. The wisdom, the understanding of God, the will of God, the immensity of God, is in that person, not as that person, but as the person is God. The like is to be said of the persons of the Son and of the Holy Ghost. Hereby each person having the understanding, the will, and power of God, becomes a distinct principle of operation; and yet all their acting ad extra being the acting of God, they are undivided, and are all the works of one, of the selfsame God. And these things do not only necessarily follow, but are directly included, in the revelation made concerning God and his subsistence in the Scriptures.

A Brief Declaration and Vindication of the Doctrine of the Trinity.

In short, simplicity is necessary to protect the doctrine of the Trinity, because it prevents a collapse of God’s oneness into some lesser threeness. To solve the “problem” of three-ness, we need not carve up God but rather understand that the Divine Essence is relational in this manner. While our language and comprehension force us to consider the matter of substance and relation separately; we must not draw the invalid conclusion that substance and relation are separate in God. Our linguistic and intellectual limitations are not limitations in God.